Longitude lrg

Grunt Productions 2005

Longitude and Spherical Longitude and Spherical TrianglesTriangles

A Brief By Lance Grindley


A few quick notes …A few quick notes …

• Nautical mileNautical mile– One minute of Earth’s circumference at the EquatorOne minute of Earth’s circumference at the Equator

• Earth’s circumference at the equator is 360 degreesEarth’s circumference at the equator is 360 degrees– Which is 360*60 = 21,600 minutesWhich is 360*60 = 21,600 minutes

• So, Earth’s circumference at the equator is 21,600 nautical So, Earth’s circumference at the equator is 21,600 nautical milesmiles

• One knot = one nm per hourOne knot = one nm per hour

• One nm = 1.15 land mileOne nm = 1.15 land mile


More notes …More notes …

• Every meridian is perpendicular to the equatorEvery meridian is perpendicular to the equator– Hence, the ease with which you could construct a Hence, the ease with which you could construct a

spherical triangle with two right anglesspherical triangle with two right angles

• So, if we travel along a fixed direction that is other than due So, if we travel along a fixed direction that is other than due east-west or north-south but at an angleeast-west or north-south but at an angle– The journey results in a spherical spiralThe journey results in a spherical spiral

• Also called a loxodromeAlso called a loxodrome

Q. What do a row of Bacardi bottles and a loxodrome have in common?A. Both are rum (rhumb) lines."


Measuring Angles on a Measuring Angles on a SphereSphere

• Lines on a sphere are great circlesLines on a sphere are great circles– intersection of sphere with a plane through the sphere’s intersection of sphere with a plane through the sphere’s

centercenter• Can define the angle between two lines as the angle made by Can define the angle between two lines as the angle made by

the two planes that create themthe two planes that create them– the smaller of the two possible choicesthe smaller of the two possible choices

• Important to figure out what a given map projection does to Important to figure out what a given map projection does to anglesangles


Spherical TrianglesSpherical Triangles

• Formed when arcs of three great Formed when arcs of three great circles meet in pairscircles meet in pairs

• Any two sides together greater Any two sides together greater than the third sidethan the third side

• Sum of interior angles can be Sum of interior angles can be (strictly) between 180(strictly) between 180°° and 540 and 540°°– Very small triangle will be Very small triangle will be

almost flat, so have just over almost flat, so have just over 180180°°

– Very large triangle can have Very large triangle can have almost 540almost 540°° degrees degrees

• Turns out that the area is directly Turns out that the area is directly proportional to the “angle proportional to the “angle excess” (how much more than excess” (how much more than 180180°° degrees its angles add up degrees its angles add up to)to) Triangle PAB v. triangle PCD


LunesLunes

• A wedge (shaped like an A wedge (shaped like an orange slice) made by two orange slice) made by two intersecting great circlesintersecting great circles

• Area of a lune is directly Area of a lune is directly proportional to angle dproportional to angle d°°– (d/360)*(4(d/360)*(4RR22))


Area of a Spherical TriangleArea of a Spherical Triangle

• For each angle d, we can consider the lune that contains For each angle d, we can consider the lune that contains that anglethat angle– The lune is the triangle + another surfaceThe lune is the triangle + another surface– The area of “triangle + another surface” equals d * The area of “triangle + another surface” equals d *

44RR22/360/360– So the (area of the three other surfaces + 3 times the So the (area of the three other surfaces + 3 times the

area of the triangle) = (sum of the angles) * 4area of the triangle) = (sum of the angles) * 4RR22/360/360– The area of the 3 other surfaces and the triangle is a The area of the 3 other surfaces and the triangle is a

hemisphere = 2hemisphere = 2RR22 = 180*(4 = 180*(4RR22/360)/360)• So area of triangle isSo area of triangle is(1/2)*(sum of angles – 180)*4(1/2)*(sum of angles – 180)*4RR22/360/360• Manifestation of how much “curvature” is captured within Manifestation of how much “curvature” is captured within

the trianglethe triangle


Angles determine a lot about Angles determine a lot about triangles on a spheretriangles on a sphere

• Two spherical triangles with the same angle sum have the Two spherical triangles with the same angle sum have the same areasame area– totally different from plane situation where all triangles totally different from plane situation where all triangles

have 180have 180°°

• If two spherical triangles have the same angles, then If two spherical triangles have the same angles, then they’re not just similar...they’re not just similar...– they have the same side lengthsthey have the same side lengths– they’re congruent!they’re congruent!


There are no ideal mapsThere are no ideal maps

• An ideal map from the sphere to the plane would preserve An ideal map from the sphere to the plane would preserve both geodesics and anglesboth geodesics and angles

• What would it do to a spherical triangle?What would it do to a spherical triangle?– Would take great circles (geodesics on sphere) to Would take great circles (geodesics on sphere) to

straight lines (geodesics on plane)straight lines (geodesics on plane)– So it would take a spherical triangle to a plane triangle, So it would take a spherical triangle to a plane triangle,

preserving all the anglespreserving all the angles– But plane triangle has 180But plane triangle has 180°° and spherical has > 180 and spherical has > 180°°!!

• The sphere is curved, and any triangle captures some of The sphere is curved, and any triangle captures some of thatthat– thus, cannot be flattened out totallythus, cannot be flattened out totally


The Navigation ProblemThe Navigation Problem

• The ancient question: The ancient question: Where am I?Where am I?

• Earth coordinates: latitude Earth coordinates: latitude and longitudeand longitude

• Latitude can be Latitude can be determined by Sun angledetermined by Sun angle

• What about longitude?What about longitude?


LatitudeLatitude

• Comparatively easyComparatively easy

• Can use Eratosthenes’ methodCan use Eratosthenes’ method– measure how far off from “directly overhead” the sun is, measure how far off from “directly overhead” the sun is,

when it is at its highest point in the sky (“local solar when it is at its highest point in the sky (“local solar noon”)noon”)

• Similar techniques using other astronomical bodies Similar techniques using other astronomical bodies – Latitude = angle from horizon to North StarLatitude = angle from horizon to North Star


Stars and other constellations helped sailors to figure out their position.The red arrow is pointing to the North Star, which is

also known as Polaris.

It is all in your stars!It is all in your stars!


This is a quadrant. A sailor would see the North Star along one edge, and where the string fell would tell approximately the

ship’s latitude.

A sailor could also use this astrolabe.

Lined it up so the sun shone through one hole onto another, and the pointer would determine the latitude.


On Land One could observe On Land One could observe Natural clocksNatural clocks

• Motion of the moon against Motion of the moon against the background of the starsthe background of the stars

• Motions of the moons of Motions of the moons of JupiterJupiter

• But these were hard to But these were hard to observe from a ship, observe from a ship, although they could be although they could be observed from landobserved from land


Techniques for measuring Techniques for measuring longitudelongitude• Find some astronomical events that recur with known Find some astronomical events that recur with known

regularityregularity– Tables compiled by Galileo, Cassini of motion of moons Tables compiled by Galileo, Cassini of motion of moons

of Jupiter (lo, Europa, Ganymede, and Callisto)of Jupiter (lo, Europa, Ganymede, and Callisto)– The moons would have eclipses at regular intervalsThe moons would have eclipses at regular intervals

• Tabulate exactly what time these eclipses occurred on Tabulate exactly what time these eclipses occurred on given daysgiven days– e.g., you have a table that says something like “Io will e.g., you have a table that says something like “Io will

have an eclipse at 7:00 PM on Jan. 22 in Paris” (It’s have an eclipse at 7:00 PM on Jan. 22 in Paris” (It’s actually rather messier than that)actually rather messier than that)

– You have a clock set to local timeYou have a clock set to local time– On Jan. 22, you look through the telescope and see the On Jan. 22, you look through the telescope and see the

eclipse at 3:00 PM local timeeclipse at 3:00 PM local time– So you are 4 hours = 60So you are 4 hours = 60° west of Paris° west of Paris


Calculating the longitudeCalculating the longitude

• Use stars or the SunUse stars or the Sun

• But in addition to making observations the need to know But in addition to making observations the need to know the time for some location of known longitudethe time for some location of known longitude– local time alone is not enoughlocal time alone is not enough

• The development of the chronometerThe development of the chronometer

• To find longitude to within 0.5 degree requires a clock that To find longitude to within 0.5 degree requires a clock that loses or gains no more than 3 seconds/dayloses or gains no more than 3 seconds/day


LongitudeLongitude

• Much more challengingMuch more challenging• Requires a way to determine how Requires a way to determine how

far you are from a fixed meridianfar you are from a fixed meridian• Essentially the same question as Essentially the same question as

“what time is it in Paris when it’s “what time is it in Paris when it’s noon here?”noon here?”– Earth rotates at constant Earth rotates at constant

velocity, once around every 24 velocity, once around every 24 hourshours

– 1 hour = 3601 hour = 360°° /24 = 15 /24 = 15° ° longitude differencelongitude difference

• Thus, need to be able to tell time Thus, need to be able to tell time at your locationat your location– e.g., pendulum clock, e.g., pendulum clock,

measuring local noonmeasuring local noon


The problem of finding The problem of finding longitude at sealongitude at sea

• To the middle of the To the middle of the 1818thth century, no century, no mechanical clock mechanical clock would keep accurate would keep accurate time in a sea-tossed time in a sea-tossed shipship


Longitude ProblemLongitude Problem• No easy way to determine longitudeNo easy way to determine longitude

• On July 8, 1714 the Longitude Act established in England to On July 8, 1714 the Longitude Act established in England to solve the “longitude problem”solve the “longitude problem”


Odd SolutionsOdd Solutions

• Anchor a series of ships across the ocean that would shoot Anchor a series of ships across the ocean that would shoot off flares and guns at set timesoff flares and guns at set times

• Telepathic connection between animals on ship and those Telepathic connection between animals on ship and those ashoreashore


Calculating longitudeCalculating longitude

• RequirementsRequirements– Clock showing base Clock showing base

meridian timemeridian time

• Record base meridian time Record base meridian time when local noon (use when local noon (use sextant)sextant)

• Calculate time difference (3 Calculate time difference (3 hrs)hrs)

• Earth rotates 360 degrees in Earth rotates 360 degrees in 24 hours 24 hours – 15 degrees in one hour15 degrees in one hour

• Three hour difference is Three hour difference is equal to 3x15 degree equal to 3x15 degree difference in longitude (45 difference in longitude (45 degrees)degrees)

0o meridianlocal

meridian

3 hrs


The ChronometerThe Chronometer

• Moons of Jupiter were too hard to observe on a shipMoons of Jupiter were too hard to observe on a ship• Jupiter’s moons still used in the 1800’sJupiter’s moons still used in the 1800’s

– Chronometers fragile for land expeditionsChronometers fragile for land expeditions• If we could just set a clock to Paris local time, and carry it If we could just set a clock to Paris local time, and carry it

with us, then when we figure out local noon, we can see with us, then when we figure out local noon, we can see what time it is in Pariswhat time it is in Paris

• Hard part is the implementationHard part is the implementation– Pendulum clocks are sensitive to being jostledPendulum clocks are sensitive to being jostled– Materials expand and contract due to temperature, Materials expand and contract due to temperature,

humidity, etc.humidity, etc.


Harrison’s chronometerHarrison’s chronometer

• John Harrison (1693-1776) John Harrison (1693-1776) invented clocks that would invented clocks that would keep good time at seakeep good time at sea


Culmination of the SunCulmination of the Sun

• Set your chronometer to Set your chronometer to some known time, say some known time, say London Time, before you London Time, before you set sailset sail


Local noon vs. time zonesLocal noon vs. time zones

• Local noon is different at every longitude on the earthLocal noon is different at every longitude on the earth

• Standardize time zones so it’s the same time in a longitude Standardize time zones so it’s the same time in a longitude “region”“region”

• Set by political agreementSet by political agreement– e.g., Newfoundland is -3:30 from standarde.g., Newfoundland is -3:30 from standard– All of China is in one time zone, even though it has about All of China is in one time zone, even though it has about

6060° of longitude° of longitude


Greenwich Meridian and Greenwich Meridian and International Date LineInternational Date Line

• Greenwich Meridian (longitude Greenwich Meridian (longitude through the Royal Observatory through the Royal Observatory in Greenwich, England) chosen in Greenwich, England) chosen as the prime meridian (0as the prime meridian (0°)°)– You can be up to 180You can be up to 180° East ° East

(ahead)(ahead)– Or Or up to 180up to 180° West (behind)° West (behind)

• On the other side of the world, On the other side of the world, the the International Date Line International Date Line is is where the “discontinuity” iswhere the “discontinuity” is


Three important time-Three important time-related datesrelated dates• 17611761

– John Harrison builds a marine chronometer with error less John Harrison builds a marine chronometer with error less than 1/5than 1/5thth of a second per day. of a second per day.

• Makes measurement of longitude possible while at sea.Makes measurement of longitude possible while at sea.

• 18841884– The demands for readable railroad schedules requires The demands for readable railroad schedules requires

adoption of Standard Time and time zones. adoption of Standard Time and time zones.

• 19051905– Albert Einstein shows that time is affected by motionAlbert Einstein shows that time is affected by motion


GPS SegmentsGPS Segments

• Space Segment: the constellation of satellitesSpace Segment: the constellation of satellites

• Control Segment: control the satellitesControl Segment: control the satellites

• User Segment: users with receiversUser Segment: users with receivers


GPS OrbitsGPS Orbits


GPS PositionGPS Position

• By knowing how far one is from By knowing how far one is from three satellites one can ideally three satellites one can ideally find their 3D coordinatesfind their 3D coordinates

• To correct for clock errors one To correct for clock errors one needs to receive four satellitesneeds to receive four satellites

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Longitude lrg